INFLATION AND DEFLATION

An interesting facet of Penrose tilings is that there are infinitely many of them. This can be proven by examining a phenomenon, called inflation and deflation, that was discovered by Roger Penrose himself. Inflation essentially turns the tiling into a less complex version of itself; the size of the tiling stays constant but the prototiles themselves are larger pieces. Deflation does the opposite; it splits the tiles of the tiling into smaller shapes that form a different pattern that is more refined.

The specific transformation is described as follows by Gardner: “Imagine that every dart is cut in half and then all short edges of the original pieces are glued together. The result: a new tiling... by larger darts and kites.” Deflation is the same process but in reverse. These operators do not maintain tile boundaries but do maintain half tiles.

A way to describe this process for the Penrose tiling that uses rhombi is by substituting triangles. Recall that this type of tiling is made up of two rhombi: a thick one and a thin one, each consisting of two isosceles triangles. The deflation process consists of breaking each of these triangles down into combinations of smaller versions of these triangles. See below: the left figure has the four different triangles that make up the rhombi shapes. The right picture shows each of the four triangles split into substitutions, consisting of smaller triangles arranged together.

Triangles that comprise the rhombi (Schweber)
Triangle substitutions (Schweber)

A tiling can be deflated by performing these substitutions on each of the shapes. In fact, the starting figure does not even need to be a tiling. One could conceivably start with just one rhombus and perform these substitutions repeatedly on its two triangles until a tiling with the desired level of refinement is reached. Inflation is the same process but in reverse. When performing an inflation, one would look for patterns that match the smaller triangle substitutions seen above, and then replace this area with a single, larger triangle. The overall process looks like the one seen below; this shows several iterations of the substitution process, demonstrating how a single tile can be turned into a fairly intricate Penrose tilings after performing just seven rounds of substitutions (deflation would, of course, be this same process but in the opposite direction).

Iterations of deflation (Schweber)

Revisiting kites and darts, we can also apply the substitution concept here. The following table shows the decompositions of both the half kites and half darts, as well as that of the common sun and star patterns. Notice how decomposing a sun creates a star in the center, and vice versa.

Deflations of half kites and half darts (Wikipedia)

Inflation or deflation can be applied to a tiling infinitely many times, creating infinite new tilings. The new shapes created by both of these rules still follow the rules of Penrose tilings. This concept shows that Penrose tiling is a form of a fractal, due to its self-similarity. The figure below demonstrates this transformation for a tiling that uses kites and darts; the original tiling is in the normal lines, and the inflated version can be seen in the overlaid heavy bold lines that outline larger tiles. Notice that the original version is also just a deflation of the inflation (it is more refined).

Inflation example (Gardner, 1988)

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